I never heard about a pidyon being required for a second child if the first was by C-section.
Correct. Although it is a machlokes in the Gemora if the firstborn son was delivered via Caesarean section if the second child (assuming its a son and is born naturally) requires a pidyon. We pasken not.
Also, I believe, but may be wrong, a pidyon is not required if the bechor was delivered by C-section.
Correct. A pidyon is not made if the bechor is born via C/Section.